Mathematics – Analysis of PDEs
Scientific paper
2011-09-26
Mathematics
Analysis of PDEs
Scientific paper
An entire solution of the Allen-Cahn equation $\Delta u=f(u)$, where $f$ has exactly three zeros at $\pm 1$ and 0, is balanced and odd, e.g. $f(u)=u(u^2-1)$, is called a $2k$-ended solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$ looks like the one dimensional, heteroclinic solution. In this paper we consider the family of four ended solutions whose ends are almost parallel at $\infty$. We show that this family can be parametrized by the family of solutions of the two component Toda system. As a result we obtain the uniqueness of four ended solutions with almost parallel ends. Combining this result with the classification of connected components in the moduli space of the four ended solutions we can classify all such solutions. Thus we show that four end solutions form, up to rigid motions, a one parameter family. This family contains the saddle solution, for which the angle between the nodal lines is $\frac{\pi}{2}$ as well as solutions for which the angle between the asymptotic half lines of the nodal set is arbitrary small (almost parallel nodal sets).
Kowalczyk Michal
Liu Ya-Ying
Pacard Frank
No associations
LandOfFree
The classification of four end solutions of the Allen-Cahn equation on the plane does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The classification of four end solutions of the Allen-Cahn equation on the plane, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The classification of four end solutions of the Allen-Cahn equation on the plane will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-689675